C3 chapter 2 - functions

In this chapter you will learn what a function is, about the domain and range of functions, how to form composite functions and how to find inverse functions.
Here's a pictorial outline of the functions topic:

Sections

C3 section 2.1

You need to know that a mapping links elements (values) in one set to elements (values) in another.
If you have studied A-level module D1 (decision mathematics), you may wish to compare a mapping with a bipartite graph.

C3 section 2.2

You need to be able to: state the domain and range of a function.

You need to know the definition of a function:

• A function is a mapping in which every element of set X (the domain) is mapped to exactly one element of set Y (the range)

There is more to be said on this, so a more sophisticated treatment is offered at C3 section 2.2 extension.

Practice questions: find the domain of functions (Kahn Academy); and find the range of a function (Kahn Academy). You may wish to do this as a class quiz using some multiple choice cards

C3 Exercise 2A

• Questions similar to this exercise:

C3 Exercise 2A worked solutions:

• 
  

C3 section 2.4

You need to be able to combine two or more simple functions to make a new more complicated function.
You need to know that:

• fg(x) means do g to x first, then do f to the result.
• fg(x) is a 'composite function.'
• beware, the order you do the functions is counter intuitive: gf(x) applies f to x first, then does g to the output:

Examples:

• Given that f(x) = x² and g(x) = x- 1, find:
  1. fg(2)
  2. fg(3)
  3. fg(x)
  4. gf(2)
  5. gf(3)
  6. gf(x)

1. fg(2) = f(g(2))
   • = f(2 - 1)
   • = (2 - 1)²
   • = 1²
   • = 1
2. fg(3) = f(g(3))
   • = (3 -1)²
   • = 4
3. fg(x) = f(g(x))
   • = (x - 1)²
   • or x² - 2x + 1 if you prefer
4. gf(2) = g(f(2))
   • = g(2²)
   • = 2² - 1
   • = 3
5. gf(3) = 3² - 1 = 8
6. gf(x) = x² - 1

• Now try fg(1) and gf(1).

• What do you notice?
• When does this happen?
• Why?
• You may find if useful to graph fg(x) and gf(x).

• A nice investigation about composite functions is provided by Jonny Griffiths' Rich Interactive Starting Points for A-Level Mathematics: RISP 18:

Jonny Griffiths' RISP 18 (click for full problem)

• Teacher notes for RISP 18 are also available, but don't spoil the investigation for yourself, try it first!

Odd and even functions